Cheese is made from milk - if the milk is non-Kosher, surely the cheese must also be non-Kosher.

​ Furthermore, the decree requiring supervised milk was promulgated before the cheese decree - meaning when we were already required to monitor the milk we purchased from our non-Jewish neighbour, we were still permitted to purchase his cheese even though it was made with unsupervised non-Kosher milk!

The answer is simple enough - Our Sages' decree did not ban all milk. It was restricted exclusively to milk collected for drinking. Milk collected for making yoghurt, butter and cheese - everything that is not a liquid for drinking, was not banned.

This is not as strange as it seems. Many decrees were put on a short leash. Chazal permitted Pas Palter, bread baked by a non-Jew when it is baked for business. This does not mean breads baked by a large factory. It refers to a fellow, your non-Jewish neighbour, who has taken up the hobby of home-baking and once a week, makes a bit of money on the side by selling his breads. These are Kosher notwithstanding they've been baked in the same kitchen and oven used for his non-Kosher meals. And he need not be a vegetarian. [The Acharonim explain that the oven is Kashered each time it is heated to be used for baking]